Multi-Resolution Beta-Divergence NMF for Blind Spectral Unmixing

Blind spectral unmixing is the problem of decomposing the spectrum of a mixed signal or image into a collection of source spectra and their corresponding activations indicating the proportion of each source present in the mixed spectrum. To perform this task, nonnegative matrix factorization (NMF) based on the $\beta$-divergence, referred to as $\beta$-NMF, is a standard and state-of-the art technique. Many NMF-based methods factorize a data matrix that is the result of a resolution trade-off between two adversarial dimensions. Two instrumental examples are (1)~audio spectral unmixing for which the frequency-by-time data matrix is computed with the short-time Fourier transform and is the result of a trade-off between the frequency resolution and the temporal resolution, and (2)~blind hyperspectral unmixing for which the wavelength-by-location data matrix is a trade-off between the number of wavelengths measured and the spatial resolution. In this paper, we propose a new NMF-based method, dubbed multi-resolution $\beta$-NMF (MR-$\beta$-NMF), to address this issue by fusing the information coming from multiple data with different resolutions in order to produce a factorization with high resolutions for all the dimensions. MR-$\beta$-NMF performs a form of nonnegative joint factorization based on the $\beta$-divergence. In order to solve this problem, we propose multiplicative updates based on a majorization-minimization algorithm. We show on numerical experiments that MR-$\beta$-NMF is able to obtain high resolutions in both dimensions for two applications: the joint-factorization of two audio spectrograms, and the hyperspectral and multispectral data fusion problem.